Is (1+cot^2(x))/(cot^2(x))=sec^2(x) ?

The answer to whether (1+cot^2(x))/(cot^2(x))=sec^2(x) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove (1+cot^2(x))/(cot^2(x))=sec^2(x)

Solution

prove

\frac{1 + cot ^{2} ( x )}{cot ^{2} ( x )} = sec^{2} (x)

Solution

True

Solution steps

\frac{1 + cot ^{2} ( x )}{cot ^{2} ( x )} = sec^{2} (x)

Manipulating left side

\frac{1 + cot ^{2} ( x )}{cot ^{2} ( x )}

Express with sin, cos

\frac{1 + cot ^{2} ( x )}{cot ^{2} ( x )}

Use the basic trigonometric identity:

cot (x) = \frac{cos ( x )}{sin ( x )}

= \frac{1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}

Simplify

\frac{1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}} : \frac{sin ^{2} ( x ) + cos ^{2} ( x )}{cos ^{2} ( x )}

\frac{1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}

(\frac{cos ( x )}{sin ( x )})^{2} = \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

(\frac{cos ( x )}{sin ( x )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

= \frac{1 + ( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{\frac{c o s ^{2} ( x )}{s i n ^{2} ( x )}}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 + \frac{c o s ^{2} ( x )}{s i n ^{2} ( x )}}{\frac{c o s ^{2} ( x )}{s i n ^{2} ( x )}}

Apply the fraction rule:

\frac{a}{\frac{b}{c}} = \frac{a \cdot c}{b}

= \frac{( 1 + \frac{c o s ^{2} ( x )}{s i n ^{2} ( x )} ) sin ^{2} ( x )}{cos ^{2} ( x )}

Join

1 + \frac{cos ^{2} ( x )}{sin ^{2} ( x )} : \frac{sin ^{2} ( x ) + cos ^{2} ( x )}{sin ^{2} ( x )}

1 + \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

Convert element to fraction:

1 = \frac{1 sin ^{2} ( x )}{sin ^{2} ( x )}

= \frac{1 \cdot sin ^{2} ( x )}{sin ^{2} ( x )} + \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot sin ^{2} ( x ) + cos ^{2} ( x )}{sin ^{2} ( x )}

Multiply:

1 \cdot sin^{2} (x) = sin^{2} (x)

= \frac{sin ^{2} ( x ) + cos ^{2} ( x )}{sin ^{2} ( x )}

= \frac{\frac{s i n ^{2} ( x ) + c o s ^{2} ( x )}{s i n ^{2} ( x )} sin ^{2} ( x )}{cos ^{2} ( x )}

Multiply

\frac{sin ^{2} ( x ) + cos ^{2} ( x )}{sin ^{2} ( x )} sin^{2} (x) : sin^{2} (x) + cos^{2} (x)

\frac{sin ^{2} ( x ) + cos ^{2} ( x )}{sin ^{2} ( x )} sin^{2} (x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( sin ^{2} ( x ) + cos ^{2} ( x ) ) sin ^{2} ( x )}{sin ^{2} ( x )}

Cancel the common factor:

sin^{2} (x)

= sin^{2} (x) + cos^{2} (x)

= \frac{sin ^{2} ( x ) + cos ^{2} ( x )}{cos ^{2} ( x )}

= \frac{sin ^{2} ( x ) + cos ^{2} ( x )}{cos ^{2} ( x )}

= \frac{cos ^{2} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x )}

Rewrite using trig identities

\frac{cos ^{2} ( x ) + sin ^{2} ( x )}{cos ^{2} ( x )}

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

= \frac{1}{cos ^{2} ( x )}

= \frac{1}{cos ^{2} ( x )}

Rewrite using trig identities

Use the basic trigonometric identity:

cos (x) = \frac{1}{sec ( x )}

\frac{1}{( \frac{1}{s e c ( x )} ) ^{2}}

Simplify

\frac{1}{( \frac{1}{s e c ( x )} ) ^{2}}

(\frac{1}{sec ( x )})^{2} = \frac{1}{sec ^{2} ( x )}

(\frac{1}{sec ( x )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{sec ^{2} ( x )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{sec ^{2} ( x )}

= \frac{1}{\frac{1}{s e c ^{2} ( x )}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{sec ^{2} ( x )}{1}

Apply rule

\frac{a}{1} = a

= sec^{2} (x)

sec^{2} (x)

sec^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

Popular Examples

prove sec^2(x)-tan^2(x)=sec(x)cos(x)prove

sec^{2} (x) - tan^{2} (x) = sec (x) cos (x)

prove cos^2(4x)= 1/2+1/2 cos(16x)prove

cos^{2} (4 x) = \frac{1}{2} + \frac{1}{2} cos (16 x)

prove-3sin(35)-3sin(15)=-6sin(25)cos(10)prove

- 3 sin (35) - 3 sin (15) = - 6 sin (25) cos (10)

prove ((sec^2(t)))/(sec^2(t)-1)=csc^2(t)prove

\frac{( sec ^{2} ( t ) )}{sec ^{2} ( t ) - 1} = csc^{2} (t)

prove sin^2(x)=cos(2x)+2 prove

sin^{2} (x) = cos (2 x) + 2

Frequently Asked Questions (FAQ)

Is (1+cot^2(x))/(cot^2(x))=sec^2(x) ?
The answer to whether (1+cot^2(x))/(cot^2(x))=sec^2(x) is True